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Search: id:A143335
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| A143335 |
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Vector matrix Markov sequence of Characteristic Polynomial: x^10 + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1: M={{0, 0, 0, 0, 0, 0, 0, 0, 0, -1}, {1, 0, 0, 0, 0, 0, 0, 0, 0, -1}, {0, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 0, 0, 1}, {0, 0, 0, 1, 0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 1, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 0, 1, 0, 0, 0, 1}, {0, 0, 0, 0, 0, 0, 1, 0, 0, 1}, {0, 0, 0, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 1, -1}}. |
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+0
2
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| 1, -1, 1, -2, 1, -2, 0, -1, -3, 2, -6, 1, -4, -3, -3, -5, -4, -7, -6, -9, -8, -14, -10, -18, -18, -20, -28, -27, -38, -39, -50, -57, -67, -79, -94, -109, -128, -154, -175, -213, -244, -292, -341, -400, -475, -553, -655, -768, -905, -1062, -1253
(list; table; graph; listen)
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OFFSET
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1,4
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COMMENT
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The sequence A142155 uses the wrong polynomial,
but is the one in http://mathworld.wolfram.com/Polylogarithm.html:
A125950 uses the wrong starting
vector as all ones.
The difference between the result here and A029826
is unexplained?
Conjecture: the similarity to the Warren Weaver ( Shannon's co-author)
polynomial for the 4 symbol telegraphic Morse code makes me wonder
if a four symbol minimal code could be based on this polynomial?
The Weaver polynomial is:
Expand[x^10*Det[{{-1, (1/x^4 + 1/x^2)}, {(1/x^6 + 1/x^5), 1/x^2 + 1/x^4 - 1}}]]=
-1 - x - x^2 - x^3 - x^6 - x^8 + x^10.
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FORMULA
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M={{0, 0, 0, 0, 0, 0, 0, 0, 0, -1}, {1, 0, 0, 0, 0, 0, 0, 0, 0, -1}, {0, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 0, 0, 1}, {0, 0, 0, 1, 0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 1, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 0, 1, 0, 0, 0, 1}, {0, 0, 0, 0, 0, 0, 1, 0, 0, 1}, {0, 0, 0, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 1, -1}}; v(0)=A029826(n),n{1,10); v(n)=M.v(n-1); a(n)=v(n)[[1]].
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MATHEMATICA
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(* generate matrix using A087612*) CompanionMatrix[p_, x_] := Module[{cl = CoefficientList[p, x], deg, m}, cl = Drop[cl/Last[cl], -1]; deg = Length[cl]; If[deg == 1, {-cl}, m = RotateLeft[IdentityMatrix[deg]]; m[[ -1]] = -cl; Transpose[m]]]; M = CompanionMatrix[x^10 + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1, x]; (* generate starting vector :v(0)=A029826(n), n{1, 10); *) f[x_] = x^10 + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1; g[x] = ExpandAll[x^10*f[1/x]]; a = Table[SeriesCoefficient[Series[1/g[x], {x, 0, 30}], n], {n, 0, 30}]; v[0] = Table[a[[n]], {n, 1, 10}]; v[n_] := v[n] = M.v[n - 1]; Table[v[n][[1]], {n, 0, 50}]
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CROSSREFS
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Cf. A087612, A125950, A029826, A070178.
Sequence in context: A028933 A143352 A127170 this_sequence A143365 A099505 A156837
Adjacent sequences: A143332 A143333 A143334 this_sequence A143336 A143337 A143338
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KEYWORD
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sign,uned,tabl,probation
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AUTHOR
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Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Oct 22 2008
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EXTENSIONS
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Appears to be identical to A143365. Presumably one or both is incorrect. - N. J. A. Sloane (njas(AT)research.att.com), Oct 25 2008
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